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It is given that vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}, \mathbf{b} = \begin{pmatrix} 5 \\ 0 \\ 1 \end{pmatrix},
\mathbf{c} = \begin{pmatrix} -1 \\ 1 \\ z \end{pmatrix}$ and $\mathbf{d} = \begin{pmatrix} k+2 \\ k-1 \\ k \end{pmatrix}$ where $z$ and $k$ are constants.
Find the angle between $\mathbf{a}$ and $\mathbf{b}$.
$84.0^\circ$
Given that $\mathbf{a}$ is perpendicular to $\mathbf{c}$, find $z$.
$z=\frac{1}{3}$
Show that $\mathbf{a}$ is perpendicular to $\mathbf{d}$ for all $k \in \mathbb{R}$.
Points $A$ and $B$ have coordinates $A(1,2,-3)$ and $B(5,0,1)$.
Find a vector $\mathbf{n}$, such that $\mathbf{n}$ is perpendiucular to $\overrightarrow{OA}$ and $\overrightarrow{OB}.$
$\mathbf{n} = \mathbf{i}-8\mathbf{j}-5\mathbf{k}$
Find the area of triangle $OAB$.
$3 \sqrt{10}$
Points $A$ and $B$ have coordinates $A(-1,2,3)$ and $B(5,0,1)$.
Remark: $A$ has changed compared to the earlier questions.
Find the length of projection of $\overrightarrow{OA}$ onto $\overrightarrow{OB}$.
$\frac{2}{\sqrt{26}}$
Find the projection vector of $\overrightarrow{OA}$ onto $\overrightarrow{OB}$.
$= -\frac{1}{13} (5\mathbf{i}+\mathbf{k})$
Find the perpendicular length from point $B$ to the line $OA$.
$= \frac{6\sqrt{35}}{7}$
It is given that $\mathbf{a} = -\mathbf{i} + 2 \mathbf{j} + 3 \mathbf{k}$.
Find the cosines of the angle $\mathbf{a}$ makes with the $x$-, $y$- and $z$-
axis respectively.
$ -\frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}} $
It is given that the direction cosines of a vector $\mathbf{b}$ with respect to the $x$ -and $y$- axes are $\frac{2}{\sqrt{6}}$ and $\frac{-1}{\sqrt{6}}$ respectively.
Find two possible vectors $\mathbf{b}$.
$2\mathbf{i}-\mathbf{j}+\mathbf{k}$ or $2\mathbf{i}-\mathbf{j}-\mathbf{k}$
Remark: For the following questions, do not use $\mathbf{a}$ and $\mathbf{b}$ from the previous questions. Vectors $\mathbf{a}$ and $\mathbf{b}$ are now unknown.
With respect to the origin $O$, points $A,B$ and $C$ have position vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}=3\mathbf{a}+\mathbf{b}$ respectively.
Find the area of triangle $OBC$ in the form $k | \mathbf{a} \times \mathbf{b} |$, where $k$ is a constant to be determined.
$ \frac{3}{2} |\mathbf{a} \times \mathbf{b}|$
It is further given that $|\mathbf{a}| = 2, \angle AOB = 60^\circ$ and $\mathbf{b}$ is a unit vector.
Find the length of projection of $\overrightarrow{OC}$ onto $\overrightarrow{OB}$.
4
Points $A,B$ and $C$ have position vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ respectively.
Give a geometrical interpretation of $|\mathbf{a} \cdot \hat{\mathbf{b}} |$.
If $c$ is a unit vector, give a geometrical interpretation of $|\mathbf{a} \times \mathbf{c}|$.
If $\mathbf{a} \cdot \mathbf{b} = 0$, what can we say about $\mathbf{a}$ and/or $\mathbf{b}$?
If $\mathbf{a} \times \mathbf{b} = \mathbf{0}$, what can we say about $\mathbf{a}$ and/or $\mathbf{b}$?